The area of a circle is 15400 cm2. What is the positive difference between the radius and the circumference of the circle? [Use π = \(\frac{22}{7}\)]
370 cm
This problem involves calculating the radius and circumference of a circle given its area and then finding the difference between these two values. We are provided with the area of the circle and the value of \(\pi\) to use.
Understanding the Circle Problem
We are given:
- Area of the circle = 15400 cm\(^2\)
- Value of \(\pi = \frac{22}{7}\)
We need to find the positive difference between the radius (\(r\)) and the circumference (\(C\)) of the circle. This difference is \(|C - r|\).
Steps to Solve the Circle Problem
Here's how we can solve this problem step-by-step:
- Use the formula for the area of a circle to find the radius (\(r\)).
- Use the formula for the circumference of a circle to find the circumference (\(C\)).
- Calculate the positive difference between the circumference and the radius.
Calculating the Radius of the Circle
The formula for the area of a circle is: Area = \(\pi r^2\)
We are given the Area = 15400 cm\(^2\) and \(\pi = \frac{22}{7}\). Let's substitute these values into the formula:
\[ 15400 = \frac{22}{7} \times r^2 \]To find \(r^2\), we can rearrange the equation:
\[ r^2 = \frac{15400 \times 7}{22} \] \[ r^2 = \frac{15400}{22} \times 7 \]We can simplify the division:
\[ \frac{15400}{22} = \frac{1540 \times 10}{22} = \frac{154 \times 100}{22} \]Since \(154 = 22 \times 7\), we have:
\[ \frac{22 \times 7 \times 100}{22} = 7 \times 100 = 700 \]So, \(r^2 = 700 \times 7\)
\[ r^2 = 4900 \]Now, we find the radius \(r\) by taking the square root of \(r^2\):
\[ r = \sqrt{4900} \]Since \(4900 = 49 \times 100 = 7^2 \times 10^2 = (7 \times 10)^2 = 70^2\), we get:
\[ r = 70 \text{ cm} \]The radius of the circle is 70 cm.
Calculating the Circumference of the Circle
The formula for the circumference of a circle is: Circumference (\(C\)) = \(2 \pi r\)
We have the radius \(r = 70\) cm and \(\pi = \frac{22}{7}\). Let's substitute these values:
\[ C = 2 \times \frac{22}{7} \times 70 \]We can simplify the expression:
\[ C = 2 \times 22 \times \frac{70}{7} \] \[ C = 2 \times 22 \times 10 \] \[ C = 44 \times 10 \] \[ C = 440 \text{ cm} \]The circumference of the circle is 440 cm.
Finding the Difference
We need to find the positive difference between the circumference and the radius. This is \(|C - r|\).
Difference = Circumference - Radius
Difference = 440 cm - 70 cm
Difference = 370 cm
The positive difference between the radius and the circumference is 370 cm.
Summary of Calculations
- Radius (\(r\)): 70 cm
- Circumference (\(C\)): 440 cm
- Positive difference: \(|440 - 70| = 370\) cm
Revision Table
| Concept | Formula | Value Used | Result |
|---|---|---|---|
| Area of Circle | \(\pi r^2\) | 15400 cm\(^2\), \(\pi = \frac{22}{7}\) | \(r = 70\) cm |
| Circumference of Circle | \(2 \pi r\) | \(r = 70\) cm, \(\pi = \frac{22}{7}\) | \(C = 440\) cm |
| Difference (C - r) | \(|C - r|\) | \(C = 440\) cm, \(r = 70\) cm | 370 cm |
Additional Information on Circle Properties
Circles are fundamental shapes in geometry. Understanding their properties like radius, diameter, circumference, and area is crucial for various mathematical and real-world applications. Here are some key definitions:
- Radius (r): The distance from the center of the circle to any point on its boundary.
- Diameter (d): The distance across the circle passing through the center. It is twice the radius (\(d = 2r\)).
- Circumference (C): The distance around the circle's boundary. It is given by the formula \(C = 2 \pi r\) or \(C = \pi d\).
- Area (A): The amount of surface enclosed by the circle. It is given by the formula \(A = \pi r^2\) or \(A = \frac{1}{4} \pi d^2\).
- \(\pi\): A mathematical constant representing the ratio of a circle's circumference to its diameter. It is approximately 3.14159, and often approximated as \(\frac{22}{7}\) or 3.14 for calculations.
These formulas are interconnected, allowing us to calculate other properties if one property is known, as demonstrated in this problem where we used the area to find the radius and then the circumference.
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